Porous medium equations on manifolds with critical negative curvature: unbounded initial data

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan–Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.     

Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

Nonlinear wave equations of carrier type on manifolds

In this paper, we investigate the existence and uniqueness of a solution for differential equations of the carrier type on lateral boundary Σ of the cylinder Q, cf. (1). The main point is to transform this initial value problem into a differential operator equation of the type , cf. (6). The operator is defined in Section 2, and it acts in Sobolev spaces on Γ, boundary of Ω. The initial value problem (6) is investigated in Section 3 by the method of Faedo–Galerkin. Thus, we obtain the existence of a weak solution for (6), and in Section 4, we prove its uniqueness. Copyright © 2012 John Wiley & Sons, Ltd.     

Notes on Metric Independent Analysis of Classical Fields

A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n‐dimensional space are represented as differential (n ? 1)‐forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric‐invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory. Copyright © 2012 John Wiley & Sons, Ltd.

     

Effects of turbulence-chemistry interactions on auto-ignition and flame structure for n-dodecane spray combustion

The Engine Combustion Network (ECN) spray A under diesel engine conditions is investigated with a non-adiabatic 5D Flamelet Generated Manifolds (FGM) model with the consideration of detailed chemical kinetic mechanisms. The enthalpy deficit due to droplet vapourisation is considered by employing an additional controlling parameter in the FGM library. In this FGM model, β-PDF is used for the PDF integration over the control variable space. Validation results in non-reacting conditions indicate relatively good agreement between the predicted and experimental data in terms of liquid and vapour penetrations and mixture fraction spatial distribution. In reacting conditions, the effects of variance of mixture fraction and progress variable were examined. The ignition delay time and the quasi-steady flame structure are both affected by the variances. The variance of mixture fraction delays the ignition process and the variance of progress variable accelerates it. For mixture fraction, the ignition process is quicker at any stage in the case of neglecting variance. While things are more complex for progress variable, the ignition process is advanced in the case of neglecting variance at early times, but surpassed by the case of β-PDF later and until auto-ignition. When variance of mixture fraction is considered, the OH mass fraction shows a wide spatial distribution. While if not, a very thin flame is observed with a higher peak in OH, and a very large lift-off length. The variance of progress variable has little impact on the global flame structure, but makes the flame lift-off length much shorter. This study confirms the general observation, that the variance of mixture fraction is of higher importance in high temperature non-premixed combustion, however, we found that the variance of progress variable is far from negligible.     

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

For the 5-components Maxwell–Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem of invariant sets for the system, we discover a rich family of periodic solutions in explicit form.